Royal Mail is facing an anti-competition probe in France, along with TNT and Fedex. Photograph: Matthew Lloyd/Getty Images

Royal Mail has warned it faces a potentially hefty fine from French competition authorities investigating possible anti-trust violations at the group's parcel delivery subsidiary.

Shares in the postal service, which is 30% state-owned following its controversial privatisation last year, were among the biggest fallers on the FTSE 100 on Wednesday down 1.9% to 479.5p. Floated at 330p, they had climbed above 600p at one point.

The group said French competition authorities were investigating breaches of anti-trust laws "in connection with a broader investigation into alleged activities within the industry in France". Dutch postal company TNT Express and the US firm FedEx have also confirmed they are being investigated and face possible fines.

Royal Mail is caught up in the inquiry via its French parcel delivery business GLS France, a loss-making part of its otherwise profitable and fast-growing European parcel business. General Logistics Systems is based in Amsterdam and operates in 37 countries: in 2013 it made an operating profit of £108m on earnings of £1.6bn, generating 17% of Royal Mail's business.

FedEx said French authorities were alleging anti-competitive actions took place at trade association meetings, involving former general managers at Tatex, the French business it bought in 2012. TNT said the allegations dated back to 2010 and it had been cooperating with French investigators. Royal Mail declined to give details of the allegations. It said: "Given the early stage of this matter, we cannot yet determine the amount or range of potential loss; however, it is possible that it could be material."

Mike van Dulken, head of research at Accendo Markets, described Royal Mail's statement as ominous but likely to be just cautious at such an early stage. "After all the talk of the group being sold off too cheap last October, it just goes to show that even previously nationalised entities … aren't whiter than white and represent risks to investors," he said.